arXiv:1104.1773v3 [q-fin.RM] 12 Feb 2013
The Annals of Applied Probability
2013, Vol. 23, No. 1, 348–385
DOI: 10.1214/12-AAP845
c Institute of Mathematical Statistics, 2013
DEFAULT CLUSTERING IN LARGE PORTFOLIOS:
TYPICAL EVENTS
By Kay Giesecke, Konstantinos Spiliopoulos
and Richard B. Sowers
Stanford University, Brown University and University of Illinois
at Urbana, Champaign
We develop a dynamic point process model of correlated default
timing in a portfolio of firms, and analyze typical default profiles in
the limit as the size of the pool grows. In our model, a firm defaults
at a stochastic intensity that is influenced by an idiosyncratic risk
process, a systematic risk process common to all firms, and past defaults. We prove a law of large numbers for the default rate in the
pool, which describes the “typical” behavior of defaults.
1. Introduction. The financial crisis of 2007–09 has made clear the need
to better understand the diversification of risk in financial systems with
interacting entities. Prior to the crisis, the common belief was that risk had
been diversified away by using the tools of structured finance. As it turned
out, the correlation between assets was much larger than supposed. The
collapse fed on itself and created a spiral.
We study the behavior of defaults in a large portfolio of interacting firms.
We develop a dynamic point process model of correlated default timing,
and then analyze typical default profiles in the limit as the number of constituent firms grows. Our empirically motivated model incorporates two distinct sources of default clustering. First, the firms are exposed to a risk factor
process that is common to all entities in the pool. Variations in this systematic risk factor generate correlated movements in firms’ conditional default
probabilities. Das, Duffie, Kapadia and Saita [5] show that this mechanism
is responsible for a large amount of corporate default clustering in the U.S.
Second, a default has a contagious impact on the health of other firms. This
Received January 2011; revised January 2012.
AMS 2000 subject classifications. 91G40, 60H10, 60G55, 60G57, 91680.
Key words and phrases. Interacting point processes, law of large numbers, portfolio
credit risk, contagion.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2013, Vol. 23, No. 1, 348–385. This reprint differs from the original in pagination
and typographic detail.
1
2
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
impact fades away with time. Azizpour, Giesecke and Schwenkler [1] provide statistical evidence for the presence of such self-exciting effects in U.S.
corporate defaults, after controlling for the exposure of firms to systematic
risk factors.
More precisely, we assume that a firm defaults at an intensity, or conditional arrival rate, which follows a mean-reverting jump-diffusion process
that is driven by several terms. The first term, a square root diffusion, represents an independent, firm-specific source of risk. The second term is a
systematic risk factor that influences all firms, and that generates diffusive
correlation between the intensities. For simplicity, we take this systematic
risk factor to be an Ornstein–Uhlenbeck process. The third term affecting
the intensity is the default rate in the pool. Defaults cause jumps in the intensity; they are common to all surviving firms. We thus have two sources of
correlation between the firms: the dependence on the systematic risk factor
and the influence of past defaults. While this formulation parsimoniously
captures several of the sources of default correlation identified in empirical
research, the intricate event dependence structure presents a challenge for
the mathematical analysis of the system.
Our goal is to understand the behavior of the default rate in the portfolio in the limit as the number of firms in the pool grows. Large stochastic
systems often tend to have macroscopic organization due to limit theorems
such as the law of large numbers. This allows us to identify typical behavior.
Our main result is a law of large numbers for the default rate in the pool;
this describes the macroscopically typical profile. The limiting default rate
satisfies an integral equation that makes explicit the role of the contagion exposure for the behavior of default clustering in the pool. The result depends
heavily on the analysis of Markov processes via the martingale problem; see
Ethier and Kurtz [11]. We will have more to say on the mathematical aspects of this in a moment. Once the typical behavior has been identified,
one can then study Gaussian fluctuations and the structure of atypically
large default clusters in the portfolio. We plan to pursue these directions in
a future work.
Previous studies have analyzed the behavior of defaults in large portfolios. Dembo, Deuschel and Duffie [9] examine a doubly-stochastic model of
default timing. In their model, default correlation is due to the exposure of
firms to a common systematic risk factor which is represented by a random
variable. Conditional on this variable, defaults are independent. A large deviation argument leads to an approximation of the tail of the conditional
portfolio loss distribution. Glasserman, Kang and Shahabuddin [14] study
a copula model of default timing using large deviation techniques. In that
formulation, default events are conditionally independent given a set of common risk factors. Bush, Hambly, Haworth, Jin and Reisinger [2] prove a law
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
3
of large numbers for a related dynamic model. Davis and Rodriguez [6] develop a law of large numbers and a central limit theorem for the default
rate in a stochastic network setting, in which firms default independently of
one another conditional on the realization of a systematic factor governed
by a finite state Markov chain. Sircar and Zariphopoulou [20] examine large
portfolio asymptotics for utility indifference valuation of securities exposed
to the losses in the pool. As with these papers, our model includes exposure
to a common systematic risk factor. In contrast, however, our model captures
the self-exciting nature of defaults. Therefore, the firms in the pool are correlated even after conditioning on the path of the systematic factor process.
The use of interacting particle systems to study the behavior of default
clustering in large portfolios is a growing area. In a mean-field model, Dai
Pra, Runggaldier, Sartori and Tolotti [3] and Dai Pra and Tolotti [4] take
the intensity of a constituent firm as a deterministic function of the percentage portfolio loss due to defaults. In a model with local interaction, Giesecke
and Weber [13] take the intensity of a constituent firm as a deterministic
function of the state of the firms in a specified neighborhood of that firm.
The interacting particle perspective leads to the study of the convergence
of interacting Markov processes, laws of large numbers for the percentage
portfolio loss, and Gaussian approximations to the portfolio loss distribution
based on central limit theorems. The interacting particle system which we
propose and study incorporates an additional source of clustering, namely,
the exposure of a firm to a systematic risk factor process. Moreover, firmspecific sources of default risk are present in our system. Also, the nature of
mean-field interaction in our system is different. In [3] and [4], a constituent
intensity is a function of the current default rate in the pool. In that formulation, the impact of a default on the dynamics of the surviving firms is
permanent. In our work, a constituent intensity depends on the path of the
default rate. The impact of a default on the surviving firms is transient, and
fades away exponentially with time. There is a recovery effect.
As we were finishing this work, we learned of a related law of large numbers
type result by Cvitanić, Ma and Zhang [16]. They take the intensity of a
constituent firm as a function of a firm-specific risk factor, a systematic risk
factor and the percentage portfolio loss due to defaults. The risk factors
follow diffusion processes whose coefficients may depend on the portfolio
loss. Our model of the risk factors is more specific than theirs, and thus we
are able to arrive at slightly more explicit results. Moreover, the effect of
defaults in [16] is permanent, as in [3] and [4].
There are several mathematical contributions in our efforts. Our analysis
of typical events (a weak convergence result) is somewhat similar to that of
certain genetic models (most notably the Fleming–Viot process; see Chapter 10 of [11], Fleming and Viot [12] and Dawson and Hochberg [7]), but the
specific form of our intensity processes imply both complications and simpli-
4
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
fications. Our work is centered on a jump-diffusion intensity process which is
driven by Ornstein–Uhlenbeck and square root diffusion terms. This formulation allows some explicit simplifying calculations which are not available in
a more abstract framework. On the other hand, due to the square root singularity, certain technical estimates need to be developed from scratch (see
Section 10). A final point of interest is heterogeneity. Interacting particle
systems are often assumed to have homogeneous dynamics, where various
parameters are the same for each particle. This allows the main mathematical arguments to take their simplest form. Practitioners in credit risk,
however, in reality face an extra problem in data aggregation, where each
firm in a portfolio has its own statistical parameters. We have framed our
weak convergence result to allow for a distribution of “types,” that is, a frequency count of the different model parameters. This leads us to the correct
effective dynamics of the portfolio and, in particular, to a precise formulation
of the effects of self-excitation (see Remark 5.2).
The rest of this paper is organized as follows. Section 2 formulates our
model of default timing. We establish that our model is well-posed via the
results of Section 3. In Section 4 we identify the limit as the number of firms
in the portfolio goes to infinity—a law of large numbers result. The proof
of this result is in Section 8, but depends upon the technical calculations of
Sections 5, 6 and 7. Section 9 concludes and discusses extensions. Section 10
contains a number of technical results on square-root-like processes which
are used in our calculations.
2. Model, assumptions and notation. We construct a point process model
of correlated default timing in a portfolio of firms. We assume that (Ω, F , P)
is an underlying probability triple on which all random variables are defined.
Let {W n }n∈N be a countable collection of standard Brownian motions. Let
{en }n∈N be an i.i.d. collection of standard exponential random variables.
Finally, let V be a standard Brownian motion which is independent of the
W n ’s and en ’s. Each W n will represent a source of risk which is idiosyncratic to a specific firm. Each en will represent a normalized default time for
a specific firm. The process V will drive a systematic risk factor process to
which all firms are exposed.
Fix an N ∈ N, n ∈ {1, 2, . . . , N } and consider the following system:
q
N,n
N,n
dWtn
dλt = −αN,n (λt − λ̄N,n ) dt + σN,n λN,n
t
N,n
C
S
+ βN,n
dLN
dXt ,
t + εN βN,n λt
(2.1)
λN,n
= λ◦,N,n ,
0
dXt = −γXt dt + dVt ,
t > 0,
t > 0,
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
5
X0 = x◦ ,
LN
t =
Z t
N
1 X
χ[en,∞)
λN,n
ds
.
s
N
s=0
n=1
def
C ∈ R = [0, ∞) and β S
Here, βN,n
+
N,n ∈ R are constants which represent the
exposure of the nth firm in the pool to LN and X, respectively. The αN,n ’s,
λ̄N,n ’s and σN,n ’s are in R+ and characterize the dynamics of the firms.
We will address the role of εN in a moment. The initial condition x◦ of
X is fixed and γ > 0. We use χ to represent the indicator function here
and throughout the paper. The description of LN is equivalent to a more
standard construction. In particular, define
Z t
def
ds
≥
e
λN,n
τ N,n = inf t ≥ 0 :
(2.2)
n .
s
s=0
Then
χ[en,∞)
(2.3)
and, consequently,
Z
LN
t
t
s=0
λN,n
ds
s
= χ{τ N,n ≤t}
N
1 X
=
χ{τ N,n ≤t} .
N
n=1
λN,n
The process
represents the intensity, or conditional event rate, of the
nth firm in a portfolio of N firms. More precisely, λN,n is the instantaneous
Doob–Meyer compensator to the default indicator process (2.3); see (4.1).
We will see in Proposition 3.3 in Section 3 that the λN,n ’s are indeed nonnegative. The process X represents a source of systematic risk; in our model
this is a stable Ornstein–Uhlenbeck process. The process LN is the default
rate in the pool. The jump-diffusion model for λN,n captures several sources
of default clustering. A firm’s intensity is driven by an idiosyncratic source
of risk represented by a Brownian motion W n , and a source of systematic
risk common to all firms—the process X. Movements in X cause correlated
changes in firms’ intensities and thus provide a source of default clustering
emphasized by [5] for corporate defaults in the U.S. The sensitivity of λN,n
S . The second source
to changes in X is measured by the parameter βN,n
C dLN .
of default clustering is through the feedback (“contagion”) term βN,n
t
C
A default causes an upward jump of size N1 βN,n
in the intensity λN,n . Due
to the mean-reversion of λN,n , the impact of a default fades away with time,
exponentially with rate αN,n . Self-exciting effects of this type have been
found to be an important source of the clustering of defaults in the U.S.,
over and above any clustering caused by the exposure of firms to systematic
risk factors [1].
6
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
C = β S = 0 for all n ∈ {1, 2, . . . , N }, the intenIn the special case that βN,n
N,n
sities λN,n follow independent square root processes so firms default independently of one another. The formulation (2.1) is a natural generalization
of the widely used square root model to address the clustering between defaults.
The interest in large pools of assets is that they provide diversification;
they allow one to construct portfolios which have small variance. The dynamics of X imply that X is stochastically of order 1, that is, it is stable.1
Thus, the only way for the pool to have small variance in our model is for
each of the constituent firms to have small exposure to X. We thus assume
that
lim εN = 0.
N →∞
If εN is not small, the influence of the systematic risk factor X will be of
order 1, and the “typical” behavior of the pool will strongly depend on X
(and the tail behavior of the whole system will be strongly determined by
the tail of X).
Remark 2.1. Given the simple structure of X, our model is equivalent,
if x◦ = 0, to a model where each intensity has exposure of order 1 to a small
systematic risk. Namely, if x◦ = 0, then εN X = X̃ N where
dX̃tN = −γ X̃tN dt + εN dVt .
Our model allows for a significant amount of bottom-up heterogeneity; the
intensity dynamics of each firm can be different. We capture these different
dynamics by defining the “types”
def
C
S
, βN,n
);
pN,n = (αN,n , λ̄N,n , σN,n , βN,n
(2.4)
def
the pN,n ’s take values in parameter space P = R4+ × R. In order to expect
regular macroscopic behavior of LN as N → ∞, the pN,n ’s and the λ◦,N,n ’s
should have enough regularity as N → ∞. For each N ∈ N, define
π
N
1 X
δpN,n
=
N
N def
n=1
and
def
ΛN
◦ =
N
1 X
δλ◦,N,n ;
N
n=1
these are elements of P(P) and P(R+ ), respectively.2
We need two main assumptions. First, we assume that the types of (2.4)
and the initial distributions (the λ◦,N,n ’s) are sufficiently regular.
1
Regulatory agencies, for example, are charged with preventing systematic factors from
spiraling out of control.
2
As usual, if E is a topological space, P(E) is the collection of Borel probability
measures on E.
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
Assumption 2.2.
7
We assume that
def
π = lim π N
N →∞
def
and Λ◦ = lim ΛN
◦
N →∞
exist [in P(P) and P(R+ ), resp.].
Note that this is what happens in practice; one constructs a frequency
count of the parameters of the different assets in a large pool and uses this
to seek aggregate dynamics for the pool itself. For a large pool, one hopes
that this frequency count will have some simpler macroscopic description.
Second, we assume that the types are bounded.
Assumption 2.3. We assume that there is a K2.3 > 0 such that the
C |’s, |β S |’s and λ
αN,n ’s, λ̄N,n ’s, σN,n ’s, |βN,n
◦,N,n ’s are all bounded by K2.3
N,n
for all N ∈ N and n ∈ {1, 2, . . . , N }.
Equivalently, we require that the πN ’s and ΛN
◦ ’s all (uniformly in N ) have
compact support. We could relax this requirement, at the cost of a much
more careful error analysis.
We are interested in the typical behavior of {LN }. In Section 3 we consider
the well-posedness of the model (2.1), while in Section 4 we state the law of
large numbers result, Theorem 4.2.
3. Well-posedness of the model. We here state several technical results
concerning the intensities which are a central part of our model. We want
to understand the structure of the λN,n ’s a bit more. The complications
which require our attention are the square root singularity, and the fact
that the λt dXt term contains the term λt Xt dt, implying that the dynamics
of the R2 -valued process (λ, X) contain a superlinear drift. The proofs of
the results here will be given in Section 10.
Let W ∗ be a reference Brownian motion with respect to a filtration
{Gt }t≥0 . Assume also that V is adapted to {Gt }t≥0 . Let ξ be a {Gt }t≥0 adapted, point process which takes values in [0, 1] and such that ξ0 = 0. Fix
p = (α, λ̄, σ, β C , β S ) ∈ P and λ◦ in R+ . Consider the SDE
p
dλt = −α(λt − λ̄) dt + σ λt ∨ 0 dWt∗ + β C dξt + β S λt dXt ,
t > 0,
(3.1)
λ0 = λ◦ .
Note that by expanding the dynamics of dX and rearranging a bit, we get
that
p
dλt = −{α + β S γXt }λt dt + αλ̄ dt + β C dξt + σ λt ∨ 0 dWt∗ + β S λt dVt .
Also, we have for the moment subsumed the small parameter εN into the
β S term; see the proof of Proposition 3.3.
We will use a number of ideas from [15] (see also [8]).
8
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
Lemma 3.1. There is a nonnegative solution λ of the R-valued SDE
(3.1). Furthermore, supt∈[0,T ] E[|λt |p ] < ∞ for all T > 0 and p ≥ 1.
We also have uniqueness.
Lemma 3.2.
The solution of (3.1) is unique.
The model (2.1) is thus well posed.
Proposition 3.3. The system (2.1) has a unique solution such that
λN,n
≥ 0 for every N ∈ N, n ∈ {1, 2, . . . , N } and t ≥ 0.
t
Proof. Using Lemmas 3.1 and 3.2, solve (2.1) between the default
times. Replace β S by εN β S in applying Lemma 3.1. We shall also need a macroscopic bound on the intensities.
Lemma 3.4.
For each p ≥ 1 and T ≥ 0,
Kp,T,3.4
N
1 X
= sup
E[|λN,n
|p ]
t
0≤t≤T N
def
N ∈N
n=1
is finite.
4. Typical events: A law of large numbers. Our first task is to understand the “typical” behavior of our system. To do so, we need to understand
a system which contains a bit more information than the default rate LN .
For each N ∈ N and n ∈ {1, 2, . . . , N }, define
Z t
def
N,n
mN,n
=
χ
(4.1)
λ
ds
= χ{τ N,n >t}
[0,en )
s
t
s=0
= 1 if and only if the nth
[where τ N,n is as in (2.2)]. In other words, mN,n
t
firm is still alive at time t; otherwise mN,n
=
0.
Thus,
mN,n is nonincreasing
t
and right-continuous. It is easy to see that
Z t
N,n
N,n
mt +
λN,n
ds
s ms
s=0
def
def
is a martingale. Define P̂ = P ×R+ . For each N ∈ N, define p̂N,n
= (pN,n , λN,n
)
t
t
for all n ∈ {1, 2, . . . , N } and t ≥ 1. For each t ≥ 0, define
def
µN
t =
N
1 X
δ N,n mN,n
t ;
N n=1 p̂t
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
9
in other words, we keep track of the empirical distribution of the type and
credit spread for those assets which are still “alive.” We note that
N
LN
t = 1 − µt (P̂)
for all t ≥ 0.
We want to understand the dynamics of µN
t for large N (this will then
).
To
understand
what our main result
imply the “typical” behavior for LN
t
is, let’s first set up a topological framework to understand convergence of
µN . Let E be the collection of sub-probability measures (i.e., defective probability measures) on P̂, that is, E consists of those Borel measures ν on P̂
such that ν(P̂) ≤ 1. We can topologize E in the usual way (by projecting
onto the one-point compactification of P̂; see [19], Chapter 9.5). In particdef
ular, fix a point ⋆ that is not in P̂ and define P̂ + = P̂ ∪ {⋆}. Give P̂ + the
standard topology; open sets are those which are open subsets of P̂ (with
its original topology) or complements in P̂ + of closed subsets of P̂ (again,
in the original topology of P̂). Define a bijection ι from E to P(P̂ + ) (the
collection of Borel probability measures on P̂ + ) by setting
def
(ιν)(A) = ν(A ∩ P̂) + (1 − ν(P̂))δ⋆ (A)
for all A ∈ B(P̂ + ). We can define the Skorohod topology on P(P̂ + ), and
define a corresponding metric on E by requiring ι to be an isometry. This
makes E a Polish space. We thus have that µN is an element3 of DE [0, ∞).
The main theorem of this section is Theorem 4.2, essentially a law of
large numbers. The construction of the limiting process will take several
steps. First, for each p = (α, λ̄, σ, β C , β S ) ∈ P, let bp satisfy
(4.2)
ḃp (t) = 1 − 21 σ 2 (bp (t))2 − αbp (t),
t > 0,
bp (0) = 0.
Note that if bp (t) = 0, then ḃp (t) = 1 > 0. Thus, bp (t) > 0 for all t > 0.
The next lemma is essential for the characterization of the limit. Its proof
is deferred to Section 10.
Lemma 4.1. There is a unique R+ -valued trajectory {Q(t); t ≥ 0} which
satisfies the equation
Z
Z t
C p
p
Q(t) =
β ḃ (t)λ +
ḃ (t − r){Q(r) + αλ̄} dr
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
3
r=0
If S is a Polish space, then DS [0, ∞) is the collection of maps from [0, ∞) into S
which are right-continuous and which have left-hand limits. The space DS [0, ∞) can be
topologized by the Skorohod metric, which we will denote by dS ; see Chapter 3.5 of [11].
10
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
× exp −b (t)λ −
(4.3)
p
× π(dp)Λ◦ (dλ).
Z
t
b (t − r){Q(r) + αλ̄} dr
p
r=0
Here, π and Λ◦ are as in Assumption 2.2.
Now let W ∗ be a reference Brownian motion. For each p̂ = (p, λ◦ ) ∈ P̂
where p = (α, λ̄, σ, β C , β S ), let λ∗t (p) be the unique solution to
Z t p
Z t
∗
∗
λ∗s (p̂) dWs∗
(λs (p̂) − λ̄) ds + σ
λt (p̂) = λ◦ − α
s=0
s=0
(4.4)
+
Z
t
Q(s) ds.
s=0
We now have our main result.
For all A ∈ B(P) and B ∈ B(R+ ), define
Z t
Z
def
∗
∗
λs (p̂) ds
χA (p)E χB (λt (p̂)) exp −
µ̄t (A × B) =
Theorem 4.2.
(4.5)
s=0
p̂=(p,λ)∈P̂
× π(dp)Λ◦ (dλ).
Then
lim P{dP(P̂ ) (µN , µ̄) ≥ δ} = 0
(4.6)
N →∞
for every δ > 0. Define
def
(4.7)
F (t) = 1 − µ̄t (P̂)
Z
=1−
p̂=(p,λ)∈P̂
Z
E exp −
t
s=0
λ∗s (p̂) ds
π(dp)Λ◦ (dλ).
Then, for all δ > 0 and T > 0,
n
o
lim P sup |LN
−
F
(t)|
≥
δ
= 0.
t
N →∞
0≤t≤T
The parts of the proof of this result will be given in Sections 5, 6 and 7.
In particular, in Section 5 we identify a candidate limit for {µN } using the
martingale problem formulation. Then in Section 6 we prove that {µN } is
tight, which ensures that the laws of {µN }’s have at least one limit point. In
Section 7 we prove that the limit is necessarily unique. Then, in Section 8
we collect things together to prove Theorem 4.2.
With this result in hand, we can rewrite (4.4) to see the exposure of a
typical firm to the contagion factor.
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
Remark 4.3.
Z
Ḟ (t) =
We have that
Z
E λ∗t (p̂) exp −
s=0
p̂=(p,λ)∈P̂
(4.8)
=
Z
t
λ∗s (p̂) ds
11
π(dp)Λ◦ (dλ)
λµ̄t (dp̂).
p̂=(p,λ)∈P̂
Thus,
(4.9)
λ∗t (p̂) = λ◦ − α
+
Z
Z
t
s=0
(λ∗s (p̂) − λ̄) ds + σ
t
Z tp
0
λ∗s (p̂) dWs∗
B(µ̄s )Ḟ (s) ds,
0
where
def
B(µ) =
Z
. Z
β λµ(dp̂)
C
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
λµ(dp̂)
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
for all µ ∈ E. In other words, the effective sensitivity of a typical intensity to the contagion is given by an average weighted by the instantaneous
intensities. Note that 0 ≤ B(µ) ≤ K2.3 .
The homogeneous case provides more explicit insights into the role of the
contagion exposure for the behavior of default clustering in the pool.
Remark 4.4. Fix p̂ = (p, λ◦ ) ∈ P̂ where p = (α, λ̄, σ, β C , β S ). Assume
that the pool is homogeneous, that is, p̂N,n = p̂ for all N ∈ N and n ∈
{1, 2, . . . , N }. By the relation (4.9), we then have that Q(t) = β C Ḟ (t). In
this case, F is given by the unique solution to the integral equation
Z t
Z t
p
C
p
p
b (t − r) dr − β
F (r)ḃ (t − r) dr − b (t)λ◦ .
F (t) = 1 − exp −αλ̄
r=0
r=0
Furthermore, if there is no exposure to contagion, that is, β C = 0, then this
integral equation reduces to the well-known explicit formula
Z t
p
p
b (t − r) dr − b (t)λ◦ .
F (t) = 1 − exp −αλ̄
r=0
Figure 1 shows the limiting default rate F (t) for different values of the contagion sensitivity β C . The default rate increases with β C . Figure 2 shows
the limiting default rate F (t) for different values of the parameter α, which
specifies the reversion speed of the intensity. The default rate is relatively
12
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
Fig. 1. Comparison of limiting default rate F (t) for different values of the contagion
sensitivity β C . The parameter case is σ = 0.9, α = 4, λ̄ = 0.5 and λ0 = 0.5.
insensitive to changes in α for shorter horizons; for longer horizons it decreases with α. The limiting default rate is more sensitive to variation in
the reversion level λ̄, as indicated in Figure 3. Variations in the diffusive
volatility σ of the intensity have little effect on F (t).
We finally note that the structure of the unperturbed (i.e., β C = β S = 0)
dynamics of the intensity (2.1) was crucial in singling out the equation
(4.3) as the proper macroscopic effect of the contagion (see the proof of
Lemma 8.2). The calculations in fact hinge upon the explicit formulae for
affine jump diffusions developed in [10]. In a more general setting we would
need a more abstract framework (see [16]).
5. Identification of the limit. We want to use the martingale problem
(see Chapter 4 of [11]) to show that µN ’s converge to a limiting process. For
every f ∈ C ∞ (P̂) and µ ∈ E, define
Z
def
f (p̂)µ(dp̂).
hf, µiE =
p̂∈P̂
Let S be the collection of elements Φ in B(P(P̂)) of the form
(5.1)
Φ(µ) = ϕ(hf1 , µiE , hf2 , µiE , . . . , hfM , µiE )
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
13
Fig. 2. Comparison of limiting default rate F (t) for different values of the reversion
speed α. The parameter case is σ = 0.9, β C = 2, λ̄ = 0.5 and λ0 = 0.5.
for some M ∈ N, some ϕ ∈ C ∞ (RM ) and some {fm }M
m=1 . Then S separates
P(P̂) (see Chapter 3.4 of [11]). It thus suffices to show convergence of the
martingale problem for functions of the form (5.1).
Let’s fix f ∈ C ∞ (P̂) and understand exactly what happens to hf, µN iE
when one of the firms defaults. Suppose that the nth firm defaults at time
t and that none of the other firms default at time t (defaults occur simultaneously with probability zero). Then
hf, µN
t iE
1
=
N
X
1≤n′ ≤N
n′ 6=n
C βN,n
′
N,n′ N,n′
+
, λt
f p
,
mN,n
t
N
′
′
1
1
′
+ f (pN,n , λN,n
)mN,n
f (pN,n , λN,n
).
t
t
t
N
N
Rt
Note, furthermore, that the default at time t means that s=0 λN,n
ds = en ,
s
N,n
so mt = 0. Hence,
hf, µN
t− iE =
(5.2)
f
N
hf, µN
t iE − hf, µt− iE = JN,n (t),
14
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
Fig. 3. Comparison of limiting default rate F (t) for different values of the reversion level
λ̄. The parameter case is σ = 0.9, β C = 2, α = 4 and λ0 = 0.5.
where
def
f
(t) =
JN,n
N C βN,n
′
1 X
N,n′ N,n′
N,n′ N,n′
+
, λt ) mN,n
− f (p
, λt
f p
t
N ′
N
n =1
1
f (pN,n , λN,n
)
t
N
for all t ≥ 0, N ∈ N and n ∈ {1, 2, . . . , N }.
We now identify the limiting martingale problem for µN . For p̂ = (p, λ)
where p = (α, λ̄, σ, β C , β S ) ∈ P and f ∈ C ∞ (P̂), define the operators
−
(5.3)
∂2f
∂f
1
(p̂) − λf (p̂),
(L1 f )(p̂) = σ 2 λ 2 (p̂) − α(λ − λ̄)
2
∂λ
∂λ
(L2 f )(p̂) =
∂f
(p̂).
∂λ
Define also
def
Q(p̂) = λβ C
for p̂ = (p, λ) where p = (α, λ̄, σ, β C , β S ) ∈ P. The generator L1 corresponds
to the diffusive part of the intensity with killing rate λ, and L2 is the macro-
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
15
scopic effect of contagion on the surviving intensities at any given time. For
Φ ∈ S of the form (5.1), define
M
X
∂ϕ
(AΦ)(µ) =
(hf1 , µiE , hf2 , µiE , . . . , hfM , µiE )
∂xm
def
m=1
(5.4)
× {hL1 fm , µiE + hQ, µiE hL2 fm , µiE }.
We claim that A will be the generator of the limiting martingale problem.
Lemma 5.1 (Weak convergence). For any Φ ∈ S and 0 ≤ r1 ≤ r2 · · · rJ =
s < t < T and {ψj }Jj=1 ⊂ B(E), we have that
#
"
Y
Z t
J
N
N
N
N
ψj (µrj ) = 0.
(AΦ)(µr ) dr
lim E Φ(µt ) − Φ(µs ) −
N →∞
r=s
j=1
Proof. For p̂ = (p, λ) where p = (α, λ̄, σ, β C , β S ) ∈ P, define
1
∂2f
∂f
(La f )(p̂) = σ 2 λ 2 (p̂) − α(λ − λ̄)
(p̂),
2
∂λ
∂λ
1 ∂2f
∂f
b
(Lx f )(p̂) = λ
(p̂) − γx
(p̂) ,
x ∈ R.
2 ∂λ2
∂λ
Then La is the generator of the idiosyncratic part of the intensity and Lbx is
the generator of the systematic risk.
We start by writing that
Z t
N
N
Φ(µt ) = Φ(µ0 ) +
{AN,1
+ AN,2
r
r } dr + Mt ,
r=0
where M is a martingale and
AN,1
=
t
M
X
∂ϕ
N
N
(hf1 , µN
t iE , hf2 , µt iE , . . . , hfM , µt iE )
∂xm
m=1
N
1 X
×
{(La fm )(p̂N,n
) + εN (LbXt fm )(p̂N,n
)}mN,n
t
t
t
N n=1
M
X
∂ϕ
N
N
(hf1 , µN
=
t iE , hf2 , µt iE , . . . , hfM , µt iE )
∂xm
m=1
b
N
× {hLa fm , µN
t iE + εN hLXt fm , µt iE },
AN,2
=
t
N
X
n=1
f1
N
λN,n
t {ϕ(hf1 , µt iE + JN,n (t),
16
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
f2
fM
N
hf2 , µN
t iE + JN,n (t), . . . , hfM , µt iE + JN,n (t))
N,n
N
N
− ϕ(hf1 , µN
.
t iE , hf2 , µt iE , . . . , hfM , µt iE )}mt
Using Lemma 3.4, it is fairly easy to see that for all f ∈ C ∞ (P̂),
Z t
N
b
|hLXr f, µr iE | dr = 0.
lim E εN
N →∞
r=0
f
. For each f ∈ C ∞ (P̂), t ≥ 0, N ∈ N and
To proceed, let’s simplify JN,n
n ∈ {1, 2, . . . , N }, define
def
f
(t) =
J˜N,n
=
Then
N
C
X
βN,n
∂f N,n N,m
(p̂ )mt
− f (pN,n , λN,n
t )
N
∂λ t
m=1
C
βN,n
hL2 f, µN
t iE
− f (p̂N
t ).
2 ∂2f f
K2.3
1
f
J (t) − J˜ (t) ≤
,
N,n
N N,n N 2 ∂λ2 C
where K2.3 is the constant from Assumption 2.3.
def
Define ι(p̂) = λ for p̂ = (p, λ) ∈ P̂. Setting
def
ÃN,2
=
t
M
X
∂ϕ
N
N
(hf1 , µN
t iE , hf2 , µt iE , . . . , hfM , µt iE )
∂xm
m=1
N
1 X N,n ˜fm
λ JN,n (t)mN,n
×
t
N n=1 t
(5.5)
M
X
∂ϕ
N
N
=
(hf1 , µN
t iE , hf2 , µt iE , . . . , hfM , µt iE )
∂xm
m=1
we have that
N
N
× {hQ, µN
t iE hL2 fm , µt iE − hιf, µt iE },
lim E
N →∞
Z
t
r=0
N,2
|AN,2
−
Ã
|
dr
= 0.
r
r
Collecting things together, we have that
"Z
Z
Z t
t
N,2
lim E
A
dr
−
AN,1
dr
+
r
r
N →∞
r=s
r=s
which implies the claim. t
r=s
(AΦ)(µN
r ) dr
Y
J
#
ψj (µN
rj )
j=1
We, in particular, note the macroscopic effect of the contagion.
= 0,
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
17
Remark 5.2. The key step in quantifying the coarse-grained effect of
contagion was (5.5). Namely, we average the combination of the jump rate
and the exposure to contagion across the pool.
6. Tightness. In this subsection we verify that the family {µN }N ∈N is
relatively compact (as a DE [0, ∞)-valued random variable); this of course is
necessary to ensure that the laws of the µN ’s have at least one limit point.
The complication of course is the feedback through contagion. We need to
show that the system is unlikely to “explode” via feedback. Our calculations
are framed by Theorem 8.6 of Chapter 3 of [11]; we need to show compact
containment and regularity of the µN ’s.
In particular, compact containment ensures that there is a compact set
K such that µN
t will belong to K for all N ∈ N and t ∈ [0, T ] with high
probability; see Lemma 6.1. Regularity shows, roughly speaking, that µN
t −
N
µs is bounded in a certain sense by a function of the time interval t − s, that
goes to zero as the length of the time interval goes to zero; see Lemma 6.3.
By Theorem 8.6 of Chapter 3 of [11], these two statements imply relative
compactness of the family {µN }N ∈N in DE [0, ∞); see Lemma 6.4.
Let’s first address compact containment.
Lemma 6.1.
E such that
For each η > 0 and t ≥ 0, there is a compact subset K of
sup P{µN
/ K} < η.
t ∈
N ∈N
0≤t<T
def
Proof. For each L > 0, define KL = [−K2.3 , K2.3 ]3 × [0, K2.3 ]2 × [0, L].
Then KL ⊂⊂ P̂, and for each t ≥ 0 and N ∈ N,
E[µN
t (P̂
N
K1,T,3.4
1 X
\ KL )] =
.
P{λN,n
≥ L} ≤
t
N n=1
L
Here K2.3 and K1,T,3.4 are the constants from Assumption 2.3 and Lemma 3.4.
Let’s next define
1
∗ def
KL = ν ∈ E : ν(P̂ \ K(L+j)2 ) < √
for all j ∈ N ;
L+j
these are compact subsets of E. We have that
∞ X
1
N
N
∗
P µt (P̂ \ K(L+j)2 ) > √
P{µt ∈
/ KL } ≤
L+j
j=1
≤
∞
X
E[µN
t (P̂ \ K(L+j)2 )]
√
1/ L + j
j=1
18
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
∞
∞
X
X K1,T,3.4
K1,T,3.4
√
≤
.
2
(L + j) / L + j j=1 (L + j)3/2
j=1
≤
Since
∞
X
K1,T,3.4
= 0,
L→∞
(L + j)3/2
j=1
lim
the result follows. We next need to understand the regularity of the µN ’s. For each t ≥ 0
and N ∈ N, we define
def
FtN = σ{λN,n
s ; 0 ≤ s ≤ t, n ∈ {1, 2, . . . , N }}.
def
Let’s also define q(x, y) = min{|x − y|, 1} for all x and y in R.
To proceed, let’s first consider the LN ’s. A useful tool will be the following
integral bound. Fix T > 0 and suppose that f is a square-integrable function
on [0, T ]. Then for any 0 ≤ s ≤ t ≤ T ,
s
Z T
Z t
√
f 2 (r) dr
f (r) dr ≤ t − s
r=0
r=s
√
Z T
1
t−s
1/4
2
≤
+ (t − s)
f (r) dr
2 (t − s)1/4
r=0
Z T
1
1/4
2
= (t − s)
f (r) dr .
1+
2
r=0
(6.1)
Define
(
)
Z t
N
N Z T
X
1
1
def 1 X
2
2
1+
(λN,n
(λN,n
1+
ΞN =
r ) dr =
r ) dr .
2N
2
N
r=0
r=0
Lemma 6.2.
n=1
n=1
Then E[ΞN ] ≤ 12 {1+K2,T,3.4 } (where K2,T,3.4 is the constant from Lemma 3.4)
and
N
N
1/4
E[|LN
E[ΞN |FsN ]
t − Ls ||Fs ] ≤ (t − s)
for all 0 ≤ s ≤ t ≤ T .
Proof. The bound E[ΞN ] ≤ 12 {1 + K2,T,3.4 } is clear from Lemma 3.4.
To proceed, let’s write
LN
t =1−
N
1 X N,n
m .
N n=1 t
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
19
N
By the martingale problem for LN , we have that LN
t = At + Mt where M
is a martingale and where
N Z
1 X t N,n N,n
N
At =
λr mr dr.
N
r=0
n=1
Thus, for 0 ≤ s ≤ t, we have (keeping in mind that LN is nondecreasing)
N
N
N
N
N
|LN
t − Ls | = Lt − Ls = At − As + Mt − Ms .
We then can use (6.1) to see that
N Z
1 X t N,n
N
λr dr ≤ (t − s)1/4 ΞN .
AN
−
A
≤
t
s
N
r=0
n=1
The claimed bound follows. Of course, P{LN
t ∈ [0, 1]} = 1 for all t ≥ 0 and N ∈ N, so compact containment (i.e., condition (a) of Theorem 7.2 of Chapter 2 of [11]) definitely
holds.
Moreover, by Lemma 6.2 we have that for any 0 ≤ t ≤ T , 0 ≤ u ≤ δ, and
0 ≤ v ≤ δ ∧ t,
N
N
N
N
N
N
N
1/4
E[q(LN
E[ΞN |FtN ].
t+u , Lt )|Ft ]q(Lt , Lt−v ) ≤ E[Lt+u − Lt |Ft ] ≤ δ
Theorem 8.6 of Chapter 3 of [11] thus implies that {LN }N ∈N is relatively
compact.
Lemma 6.3. There is a random variable HN with supN ∈N E[HN ] < ∞,
such that for any 0 ≤ t ≤ T , 0 ≤ u ≤ δ, and 0 ≤ v ≤ δ ∧ t,
N
2
N
N
N
1/4
E[q 2 (hf, µN
E[HN |FtN ].
t+u iE , hf, µt iE )q (hf, µt iE , hf, µt−v iE )|Ft ] ≤ δ
Proof. We start by using (5.2) to see that
1,N
N
hf, µN
+ A2,N
+ Bt1,N + Bt2,N ,
t iE = hf, µ0 iE + At
t
where
A1,N
t
N Z
1 X t 1,N,n
as
ds,
=
N
s=0
n=1
A2,N
t
=
Bt1,N =
N Z
X
t
n=1 s=0
f
JN,n
(s) d(1 − mN,n
s ),
q
N Z
∂f N,n
1 X t
N,n
dWsn ,
(p̂s ) λN,n
σN,n
s ms
N n=1 s=0
∂λ
20
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
Bt2,N
N Z
1 X t S N,n ∂f N,n N,n
(p̂ )ms dVs ,
βN,n λs
= εN
N
∂λ s
s=0
n=1
where, for simplicity, we have defined
2
1,N,n def 1
2
S
2 ∂ f
as
=
(p̂N,n )
(σN,n
λN,n
+ ε2N (βN,n
)2 (λN,n
s
s ) )
2
∂λ2 s
+ (−αN,n (λN,n
s
S
− λ̄N,n ) − εN βN,n
λN,n
s Xs )
∂f N,n
(p̂ ) mN,n
s .
∂λ s
Thus, for any 0 ≤ s ≤ t ≤ T ,
N
N
E[q 2 (hf, µN
t iE , hf, µs iE )|Fs ]
2,N
1,N
N
2
2,N
N
≤ 4{E[q 2 (A1,N
t , As )|Fs ] + E[q (At , As )|Fs ]
+ E[q 2 (Bt1,N , Bs1,N )|FsN ] + E[q 2 (Bt2,N , Bs2,N )|FsN ]}
2,N
N
N
− A2,N
≤ 4{E[|A1,N
− A1,N
s ||Fs ] + E[|At
s ||Fs ]
t
+ E[|Bt1,N − Bs1,N |2 |FsN ] + E[|Bt2,N − Bs2,N |2 |FsN ]}.
We now need to get some bounds. Due to Assumption 2.3, for any 0 ≤
s ≤ t < T , we have that
∂f 1
f
K2.3 |JN,n (t)| ≤
∂λ + kf k .
N
C
This implies
|A2,N
t
− A2,N
s |≤
∂f N
+ kf k |LN
K2.3 t − Ls |;
∂λ C
thus, by Lemma 6.2 we have that
E[|A2,N
t
N
1/4
− A2,N
s ||Fs ] ≤ (t − s)
∂f K2.3 + kf k E[ΞN |FsN ]
∂λ
C
for all 0 ≤ s ≤ t ≤ T . To bound the increments of A1,N , define
(
)
N Z
1 X t
(1) def 1
1,N,n 2
1+
(ar
) dr .
ΞN =
2
N
r=0
n=1
(1)
By Lemmata 3.4 and 10.1 we have that supN ∈N E[ΞN ] < ∞. By (6.1),
(1)
1/4
|A1,N
− A1,N
E[ΞN |FsN ].
s | ≤ (t − s)
t
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
21
We next turn to the martingale terms. We have that
E[|Bt1,N − Bs1,N |2 |FsN ]
#
"
2 q
N Z
1 X t
∂f N,n
N,n
dr FsN
(p̂ ) λN,n
=E
σN,n
r mr
N n=1 r=s
∂λ r
#
"
N Z
1 X t
∂f N,n 2 N,n N
≤E
(p̂ ) λr dr Fs
σN,n
N
∂λ r
r=s
n=1
(2)
≤ (t − s)1/4 E[ΞN |FsN ],
E[|Bt2,N − Bs2,N |2 |FsN ]
"Z
!2
#
N
t
X
1
∂f
S
= ε2N E
βN,n
λN,n
(p̂N,n )mN,n
dr FsN
r
r
∂λ s
r=s N n=1
#
"
2
N Z t X
∂f
1
N
2
S
(p̂N,n ) (λN,n
βN,n
≤ ε2N E
r ) dr Fs
N
∂λ r
r=s
n=1
(2)
where
≤ ε2N (t − s)1/4 E[ΞN |FsN ],
(
1
1+
2
(
(3) def 1
1+
ΞN =
2
(2) def
ΞN =
We have that
)
N Z
1 X T
∂f N,n 4 N,n 2
(p̂ ) (λr ) dr ,
σN,n
N n=1 r=0
∂λ r
)
4
N Z
1 X T
∂f
4
S
(p̂N,n ) (λN,n
βN,n
r ) dr .
N
∂λ r
r=0
n=1
(2)
sup E[ΞN ] < ∞
N ∈N
and
(3)
sup E[ΞN ] < ∞.
N ∈N
Collecting things together, we get that for any 0 ≤ t ≤ T , 0 ≤ u ≤ δ, and
0 ≤ v ≤ δ ∧ t,
N
N 2
N
N
E[q 2 (hf, µN
t+u iE , hf, µt iE )|Ft ]q (hf, µt iE , hf, µt−v iE )
N
N
≤ E[q 2 (hf, µN
t+u iE , hf, µt iE )|Ft ]
∂f (1)
(2)
1/4
2 (3) N
≤ 4δ E ΞN + K2.3 + kf k ΞN + ΞN + εN ΞN Ft .
∂λ C
We can now prove the desired relative compactness.
Lemma 6.4.
The sequence {µN }N ∈N is relatively compact in DE [0, ∞).
22
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
Proof. Given Lemmas 6.1 and 6.3, the statement follows by Theorem 8.6 of Chapter 3 of [11]. 7. Uniqueness. We next verify that the solution of the resulting martingale problem is unique. We will use a duality argument (cf. Chapter 4.4
of [11]). In particular, here duality means that existence of a solution to a
dual problem ensures uniqueness to the original problem.
Lemma 7.1 (Uniqueness). There is at most one solution of the martingale problem for A of (5.4) with initial condition π × Λ◦ .
Proof. We will use the duality arguments of Chapter 4.4 of [11]. Define
S∞
∞
M
∗
M =1 C (P̂ ). Let’s begin by defining a flow on E as follows. Fix
f ∈ E ∗ . Then f ∈ C ∞ (P̂ M ) for some M ∈ N. Fix next (p̂1 , p̂2 , . . . , p̂M ) ∈ P̂ M
C , β S ) for m ∈ {1, 2, . . . , M }.
where p̂m = (pm , λm ) and pm = (αm , λ̄m , σm , βm
m
Define
"
" M Z
##
t
X
def
∗,2
∗,M
(Tt f )(p̂1 , p̂2 , . . . , p̂M ) = E f (p̂∗,1
) exp −
λ∗,m
ds ,
s
t , p̂t , . . . , p̂t
def
E∗ =
m=1 s=0
where p̂∗,m
= (p, λ∗,m
t
t ) and
Z
Z t
∗,m
∗,m
(λs − λ̄m ) ds + σm
λt = λm − αm
t
s=0
s=0
for all m ∈ {1, 2, . . . , M }. We also define
C
(Hm f )(p̂1 , p̂2 , . . . , p̂M , p̂M +1 ) = M βM
+1 λM +1
p
λ∗,m
dWsm
s
∂f
(p̂1 , p̂2 , . . . , p̂M )
∂λm
for m ∈ {1, 2, . . . , M } and p̂M +1 = (pM +1 , λM +1 ) ∈ P̂ where pM +1 = (αM +1 ,
C
S
∗
∞
M
λ̄M +1 , σM +1 , βM
+1 , βM +1 ). Suppose that f ∈ E and that in fact f ∈ C (P̂ )
def
for some M ∈ N. Let e be an exponential(1) random variable. Set Ft = Tt f
for t < e. Select m ∈ {1, 2, . . . , M } according to a uniform distribution on
def
{1, 2, . . . , M } and set Fe = Hm (Te f ). Restart the system.
Let’s now connect F to µ. Fix f ∈ E ∗ and µ ∈ E. Then f ∈ C ∞ (P̂ M ) for
some M ∈ N, and we define
Z
def
f (p̂1 , p̂2 , . . . , p̂M )µ(dp̂1 )µ(dp̂2 ) · · · µ(dp̂M ).
(7.1) φ(µ, f ) =
(p̂1 ,p̂2 ,...,p̂M )∈P̂ M
∞
If we fix 1 = m1 < m2 < m3 < · · · < mL+1 = M + 1 and {f˜l }L
l=1 ⊂ C (P̂)
and assume that
Y
Y ˜
fl (p̂m )
f (p̂1 , p̂2 , . . . , p̂M ) =
1≤l≤L
ml ≤m<ml+1 −1
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
23
for all (p̂1 , p̂2 , . . . , p̂M ) ∈ P̂ M , then
L
Y
m
−m −1
φ(µ, f ) = hf˜l , µiE l+1 l .
l=1
By Stone–Weierstrass, we can thus approximate Φ in S by linear combinations of functions of the form φ(·, f ) of (7.1) for some f ’s in E.
To proceed, let’s fix f ∈ E and apply A to the function µ 7→ φ(µ, f ) given
by (7.1). It is fairly easy to see that if {µ̄∗t }t≥0 satisfies the martingale problem for A, then for each f ∈ E,
Z t
(1)
h1 (µ̄∗s , f ) ds + Mt ,
ϕ(µ̄∗t , f ) =
s=0
is a martingale and where, if f ∈ C ∞ (P̂ M ),
M Z
X
h1 (µ, f ) =
{(L1,m f )(p̂) + hQ, µiE (L2,m f )(p̂)}
where
M(1)
M
m=1 p̂=(p̂1 ,p̂2 ,...,p̂M )∈P
× µ(dp̂1 )µ(dp̂2 ) · · · µ(dp̂M ),
where L1,m and L2,m denote, respectively, the actions of L1 and L2 defined
by (5.3) on the mth coordinate of f . On the other hand, we also have that
for µ ∈ E,
Z t
(2)
h2 (µ, Fs ) ds + Mt ,
ϕ(µ, Ft ) =
s=0
where
M(2)
is a martingale and
M Z
X
h2 (µ, f ) =
M
m=1 p̂=(p̂1 ,p̂2 ,...,p̂M )∈P
(L1,m f )(p̂)µ(dp̂2 ) · · · µ(dp̂M )
M
1 X
+
{ϕ(µ, Hm f ) − ϕ(µ, f )}.
M
m=1
Note that
M
1 X
ϕ(µ, Hm f )
M
m=1
=
M Z
X
M +1
m=1 p̂=(p̂1 ,p̂2 ,...,p̂M ,p̂M +1 )∈P
=
M Z
X
M
m=1 p̂=(p̂1 ,p̂2 ,...,p̂M )∈P
C
βM
+1 λM +1
∂f
(p̂)µ(dp̂2 ) · · · µ(dp̂M +1 )
∂λm
hQ, µiE (L2,m f )(p̂)µ(dp̂2 ) · · · µ(dp̂M ).
24
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
Collecting things together, we have that
h1 (µ, f ) = h2 (µ, f ) + ϕ(µ, f )
and this implies uniqueness. 8. Proof of main theorem. We now have our first convergence result. Let
QN be the P-law of µN , that is,
def
QN (A) = P{µN ∈ A}
for all A ∈ B(DE [0, ∞)). Thus, QN ∈ P(DE [0, ∞)) for all N ∈ N. For ω ∈
def
DE [0, ∞), define Xt (ω) = ω(t) for all t ≥ 0.
Proposition 8.1. We have that QN converges [in the topology of
P(DE [0, ∞))] to the solution Q of the martingale problem generated by A of
(5.4) and such that QX0−1 = δπ×Λ◦ . In other words, Q{X0 = π × Λ◦ } = 1 and
for all Φ ∈ S and 0 ≤ r1 ≤ r2 ≤ · · · ≤ rJ = s < t < T and {ψj }Jj=1 ⊂ B(E),
we have that
#
"
Y
Z t
J
Q
ψj (Xrj ) = 0,
(AΦ)(Xr ) dr
lim E
Φ(Xt ) − Φ(Xs ) −
N →∞
r=s
j=1
where EQ is the expectation operator defined by Q.
Proof. The result follows from Lemmata 5.1, 6.4 and 7.1. Of course,
we also have that for any Φ ∈ S,
EQ [Φ(X0 )] = lim EQ [Φ(µN
0 )] = Φ(π × Λ◦ ),
N →∞
which implies the claimed initial condition. We next want to identify Q.
Lemma 8.2.
We have that Q = δµ̄ , where µ̄ is given by (4.5).
Proof. Recall (4.4) and the operators L1 , L2 from (5.3) and the definition of Q in (4.3). For any f ∈ C ∞ (P̂),
Z t
Z
∗
∗
λs (p̂) ds π(dp)Λ◦ (dλ).
E f (p, λt (p̂)) exp −
hf, µ̄t iE =
s=0
p̂=(p,λ)∈P̂
Thus,
d
hf, µ̄t iE =
dt
Z
p̂=(p,λ)∈P̂
E
(L1 f )(p, λ∗t (p̂)) exp
Z
−
t
s=0
λ∗s (p̂) ds
π(dp)Λ◦ (dλ)
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
+
Z
E
p̂=(p,λ)∈P̂
(L2 f )(p, λ∗t (p̂))Q(t) exp
Z
−
t
s=0
λ∗s (p̂) ds
25
× π(dp)Λ◦ (dλ)
= hL1 f, µ̄t iE + Q(t)hL2 f, µ̄tiE .
To proceed, define
Z
def
G(t) =
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
Z
β C E exp −
t
s=0
λ∗s (p̂) ds
On the one hand, we have that
Z
Z
C
∗
β E λt (p̂) exp −
Ġ(t) = −
s=0
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
=−
Z
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
t
π(dp)Λ◦ (dλ).
λ∗s (p̂) ds
π(dp)Λ◦ (dλ)
β C λµ̄t (dp̂) = −hQ, µ̄t iE .
We want to show that
(8.1)
Ġ(t) = −Q(t).
Indeed, fix p̂ = (p, λ) ∈ P̂ where p = (α, λ̄, σ, β C , β S ). Define
Z
Z t
def
bp (t − r){Q(r) + αλ̄} dr −
Ms = exp −bp (t − s)λ∗s (p̂) −
s
r=0
r=s
λ∗r (p̂) dr
for 0 ≤ s ≤ t. Using the calculations of [10],
dMs = dMs + {ḃp (t − s)λ∗s (p̂) − bp (t − s){−α(λ∗s (p̂) − λ̄) + Q(s)}
+ 21 σ 2 (bp (t − s))2 λ∗s (p̂) + bp (t − s)(Q(s) + αλ̄) − λ∗s (p̂)}Ms ds
= dMs ,
where M is a martingale [we use here the ODE (4.2)]. Noting that
Z t
p
p
b (t − r){Q(r) + αλ̄} dr ,
M0 = exp −b (t)λ −
r=0
Z
Mt = exp −
t
r=0
λ∗r (p̂) dr
we have that
G(t) =
Z
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
C
,
β exp −bp (t)λ
26
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
−
Z
t
b (t − r){Q(r) + αλ̄} dr
p
r=0
× π(dp)Λ◦ (dλ).
Differentiating this, we get that
Z
Z
Ġ(t) = −
β C ḃp (t)λ +
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
t
r=0
× exp −b (t)λ −
= −Q(t),
p
× π(dp)Λ◦ (dλ)
ḃp (t − r){Q(r) + αλ̄} dr
Z
t
b (t − r){Q(r) + αλ̄} dr
p
r=0
where we have used the defining equation (4.3) for Q. Thus, (8.1) holds, so
we have that
d
hf, µ̄t iE = hL1 f, µ̄t iE + hQ, µ̄t iE hL2 f, µ̄t iE .
dt
Thus,
Z t
(AΦ)(µ̄s ) ds,
Φ(µ̄t ) = Φ(µ̄0 ) +
s=0
and, hence, δµ̄ satisfies the martingale problem generated by A. Of course,
we also have that µ̄0 = π × Λ◦ . By uniqueness, the claim follows. We now can finish the proof of our main result.
Proof of Theorem 4.2. Since weak convergence to a constant implies convergence in probability, we have (4.6). Using the fact that the map
ϕ : P̂ 7→ 1 is in C(P̂), LN is a continuous transformation of µN into DR [0, ∞).
From (4.7) we have that
lim P{dR (LN , F ) ≥ δ} = 0
N →∞
for each δ > 0. To finish the proof, we need to replace the Skorohod norm
def
dR by the supremum norm. From (4.8) we have that KT = sup0≤t≤T Ḟ (t) is
finite for each T > 0. To get the claimed convergence, we adopt the notation
of Chapter 3.5 of [11]. For any nondecreasing and differentiable map g of
[0, T ] into itself and any t ∈ [0, T ], we have that
N
|LN
t − F (t)| ≤ |Lt − F (g(t))| + |F (g(t)) − F (t)|
≤ sup |LN
t − F (g(t))| + KT |g(t) − t|
0≤t≤T
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
27
≤ sup |LN
t − F (g(t))| + KT T sup |ġ(t) − 1|
0≤t≤T
0≤t≤T
≤ sup |LN
t − F (g(t))|
0≤t≤T
n
h
i
+ KT T max exp sup |log ġ(t)| − 1,
0≤t≤T
o
h
i
exp − sup |log ġ(t)| − 1 .
0≤t≤T
Varying g, we get that
sup |LN
t − F (t)|
0≤t≤T
≤ dR (LN , F ) + KT T max{|exp[dR (LN , F )] − 1|,
|exp[−dR (LN , F )] − 1|}.
The claim now follows; note that F and LN both take values in [0, 1]. 9. Conclusion and extensions. We have developed a point process model
of correlated default timing in a portfolio of firms, and have analyzed typical
default profiles in the limit as the size of the pool grows. Our empirically motivated model captures two important sources of default clustering, namely,
the exposure of firms to a systematic risk process, and contagion. We have
proved a law of large numbers for the default rate in the pool.
There are several potential extensions of our work. For example, the default intensity dynamics (2.1) can be generalized to include a dependence
on the systematic risk process of the magnitude of the jump at a default.
Then, the impact of a default on the surviving firms depends on the state of
the systematic risk: intuitively, if the economy is weak, firms are fragile and
more susceptible to contagion. This generalization of the intensity dynamics
is empirically plausible, and can be treated with arguments similar to the
ones we currently use.
10. Proofs of Lemmas 3.1, 3.2, 3.4 and 4.1. In this section we prove
Lemmas 3.1, 3.2, 3.4 and 4.1. For presentation purposes, we first collect in
Lemma 10.1 some a-priori bounds that will be useful in the proof of these
lemmas. Then, in Section 10.2 we proceed with the proof of Lemmas 3.1, 3.2
and 3.4. We mention here that the square-root singularity unavoidably complicates the analysis. The theory behind CIR-like processes is a bit delicate
due to the square root singularity in the diffusion, so we need to develop some
new modifications to existing results (cf. [15, 17, 18]). Last, in Section 10.3
we prove Lemma 4.1.
28
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
10.1. Effect of systematic risk. Our first step is to get some usable bounds
on the systematic risk X. We need these bounds since, as we mentioned in
Section 3, the λt dXt term contains the term λt Xt dt, implying that the dynamics of the R2 -valued process (λ, X) contain a superlinear drift. Note that
the systematic risk process X of course has an explicit form:
Z t
−γt
e−γ(t−s) dVs ,
t > 0.
Xt = e x◦ +
s=0
Fix p = (α, λ̄, σ, β C , β S ) ∈ P, λ◦ in R+ , and ξ as required in the beginning
of Section 3. Define
Z t
def
Xs ds,
Γt = αt + β S γ
def
Zt = λ◦ + αλ̄
= λ◦ + αλ̄
Z
s=0
t
e
s=0
Z t
s=0
= λ◦ +
Z
t
s=0
Γs
Γs
e
ds + β
C
Z
t
eΓs dξs
s=0
Z
ds + β eΓt ξt −
C
t
S
Γs
e ξs (α + β γXs ) ds
s=0
eΓs {αλ̄ − β C ξs (α + β S γXs )} ds + β C ξt eΓt
for all t ≥ 0. The alternate representations of Z will allow us bounds which
are independent of ξ.
Our first result is a bounds on X, Γ and Z which explicitly depend on
various coefficients. The importance of the bound on the moments of Zt is
that they do not depend on ξ.
For each p ≥ 1 and t ≥ 0,
1
(2p)! 1/(2p)
2p 1/(2p)
E[Xt ]
≤ |x◦ | + √
,
2 γ
p!
Lemma 10.1.
E[exp[pΓt ]] ≤ exp[|p|{αt + |β C x◦ |} + 21 (pβ C )2 t],
E[Zt2p ]1/(2p) ≤ λ◦ + |β C |E[e−2pΓt ]1/(2p)
1/(4p)
Z t
1−1/(2p)
−4pΓs
+t
E[e
] ds
s=0
× αλ̄t1/(4p) + α|β C |t1/(4p)
Z
+ |β β γ|
C S
t
s=0
E[Xs4p ] ds
1/(4p) .
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
Proof. We first bound X. For every p ≥ 1 and t ≥ 0
2p 1/(2p)
Z t
2p 1/(2p)
−γ(t−s)
−γt
e
dVs E[Xt ]
≤ |x◦ e | + E s=0
−γt
= |x◦ e
|+
s
Z
t
e−2γ(t−s) ds
s=0
1
≤ |x◦ | + √
2γ
(2p)!
2p p!
1/(2p)
1
= |x◦ | + √
2 γ
(2p)!
p!
1/(2p)
(2p)!
2p p!
1/(2p)
.
Next note that
Γt = αt − β S γ
S
−β γ
Z
Z
t
x◦ e−γs ds
s=0
t
s=0
Z
s
−γ(s−r)
e
dVr ds
r=0
S
−γt
= αt − β x◦ {1 − e
S
−γt
= αt − β x◦ {1 − e
Thus, for any p ∈ R
S
}−β γ
}−β
S
Z
Z
t
r=0
t
r=0
Z
t
−γ(s−r)
e
s=r
ds dVr
{1 − e−γ(t−r) } dVr .
E[exp[pΓt ]] = exp p{αt + β C x◦ (1 − e−γt )}
Z
(pβ C )2 t
−γ(t−r) 2
{1 − e
} dr
+
2
r=0
1
C 2
C
≤ exp |p|{αt + |β x◦ |} + (pβ ) t .
2
We can finally bound Z. We have that
2p 1/(2p)
Z t
2p 1/(2p)
−Γs
C
S
e
{αλ̄ − β ξs (α + β γXs )} ds
E[Zt ]
≤ λ◦ + E
s=0
C
−2pΓt 1/(2p)
+ |β |E[e
]
.
We also have that
2p 1/(2p)
Z t
Γs
C
S
e {αλ̄ − β ξs (α + β γXs )} ds
E
s=0
29
30
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
t
≤E
Z
t
≤E
Z
2Γs
ds
e
ds
s=0
Z
1−1/(2p)
×E
≤t
e
s=0
×E
≤t
2Γs
Z
p Z
S
Z
t
S
−4pΓs
e
ds
s=0
2p 1/(4p)
S
2p
{αλ̄ − β ξs (α + β γXs )} ds
t
p 1/(2p)
1/(4p)
C
Z
2
{αλ̄ − β ξs (α + β γXs )} ds
{αλ̄ − β ξs (α + β γXs )} ds
t
1−1/(2p)
s=0
C
s=0
s=0
C
2p 1/(4p)
t
E
t
−4pΓs
E[e
] ds
s=0
1/(4p)
1/(4p)
Z
1/(4p)
C 1/(4p)
C S
× αλ̄t
+ α|β |t
+ |β β γ|E
t
s=0
Xs4p ds
1/(4p) .
Combine things together to get the bound on Z. 10.2. Proofs of Lemmas 3.1, 3.2 and 3.4. Let’s next understand the regularity of various CIR-like processes which we use. Before proceeding with
the proofs, we define a function ψη (x) that will be essential for the proofs. It
is introduced in order to deal with the square-root singularity. In particular,
let
Z |x| Z y
2
1
def
def
ψη (x) =
χ 1/2 (z) dz dy and gη (x) = |x| − ψη (x)
ln η −1 y=0 z=0 z [η,η ]
for all x ∈ R.
Let us then study some important properties of ψη (x) that will be repeatedly used in the proofs. First, we note that ψη is even, so gη is also even.
Taking derivatives, we have that
Z x
1
2 1
2
χ[η,η1/2 ] (z) dz and ψ̈η (x) =
χ 1/2 (x)
ψ̇η (x) =
−1
ln η
ln η −1 x [η,η ]
z=0 z
√
for all x > 0. Since g̈η = −ψ̈η ≤ 0, ġη is nonincreasing. For x > η,
ġη (x) = 1 − 2
ln η 1/2 − ln η
= 0,
ln(1/η)
√
so in fact ġη is nonnegative on (0, ∞) and it vanishes on [ η, ∞). Thus, gη
√
is nondecreasing and reaches its maximum at η. Since gη (0) = 0, we in fact
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
31
have that
√
0 ≤ gη (x) ≤ gη ( η)
for all x ≥ 0. Since ġη is nonincreasing on (0, ∞) and ġη (x) = 1 for x ∈ (0, η),
√
√
√
we have that ġη (x) ≤ 1 for all x ∈ (0, η), so gη ( η) ≤ η. Since gη is even,
√
we in fact must have that |gη (x)| ≤ η for all x ∈ R. Hence,
√
|x| ≤ ψη (x) + η
for all x ∈ R. We finally note that
1
1 1
2
2
χ
(|x|) ≤
min
,
|ψ̈η (x)| ≤
ln η −1 |x| [η,∞)
ln η −1
|x| η
for all x ∈ R.
Now we have all the necessary tools to proceed with the proof of the
lemmas.
Proof of Lemma 3.1.
For each N ∈ N, define
def
̺N (t) =
⌊tN ⌋
N
for all t ∈ [0, T ]. For each N ∈ N, define
Z
Z t
q
∗
Γs /2
N
N
(Y̺N (s) + Zs ) ∨ 0 dWs + β
e
Yt = σ
t
s=0
s=0
((Y̺NN (s) + Zs ) ∨ 0) dVs .
We will show that (Zt + YtN )eΓt converges to a solution of (3.1) (as N ր ∞).
As a first step, let’s bound some moments. Fix p > 1. For 0 ≤ s ≤
t ≤ T , [17], Exercise 3.25, gives us that
2p Z t
N
((Y̺N (r) + Zr ) ∨ 0) dVr E r=s
≤ (p(2p − 1)) (t − s)
Z
≤ (p(2p − 1))p (t − s)p−1
Z
p
p−1
t
r=s
t
r=s
E[((Y̺NN (r) + Zr ) ∨ 0)2p dr]
E[|Y̺NN (r) + Zr |2p ] dr
≤ 22p−1 (p(2p − 1))p (t − s)p−1
Z t
Z
2p
N
E[|Y̺N (r) | ] dr +
×
r=s
t
r=s
Similarly,
2p Z t
q
∗
Γ
/2
N
r
(Y̺N (r) + Zr ) ∨ 0 dWr e
E r=s
E[|Zr | ] dr .
2p
32
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
p
p−1
≤ (p(2p − 1)) (t − s)
Z
1
≤ (p(2p − 1))p (t − s)p−1
2
t
r=s
Z
E[epΓr |(Y̺NN (r) + Zr ) ∨ 0|p ] dr
t
2pΓr
E[e
r=s
1
≤ (p(2p − 1))p (t − s)p−1
2
Z t
Z
2pΓr
2p−1
E[e
] dr + 2
×
r=s
t
+2
2p
+ Zr | ] dr
E[|Y̺NN (r) |2p ] dr
r=s
2p−1
] + E[|Y̺NN (r)
Z
t
E[|Zr | ] dr .
2p
r=s
We can bound the effect of Z by Lemma 10.1. Collecting things together,
and using the fact that ̺N (t) ≤ t, we have that there is a KA > 0 such that
Z ̺N (t)
E[|Y̺NN (s) |2p ] dr
E[|Y̺NN (t) |2p ] ≤ KA + KA
≤ KA + KA
Z
s=0
t
s=0
E[|Y̺NN (s) |2p ] dr
for all N ∈ N and t ∈ [0, T ], which in turn implies that
(10.1)
sup E[|Y̺NN (t) |2p ] ≤ KA eKA T
0≤t≤T
for 0 ≤ t ≤ T . This in turn implies that there is a KB > 0 such that
(10.2)
E[|YtN − Y̺NN (t) |2p ] ≤ KB |t − ̺N (t)|p ≤ KB
1
Np
for all 0 ≤ t ≤ T .
We next want to show that Y N converges in L1 . Fix N and N ′ in N and
define
′
def
′
νtN,N = YtN − YtN .
Fix also η > 0. We have that
′
′
′
′
√
√
+ Mt + η,
+ β 2 A2,N,N
|νtN,N | ≤ ψη (νtN,N ) + η = σ 2 A1,N,N
t
t
where M is a martingale and
Z
q
q
1 t
′
′
1,N,N ′
=
At
ψ̈η (νsN,N )eΓs { (Y̺NN (s) + Zs ) ∨ 0 − (Y̺N ′ (s) + Zs ) ∨ 0}2 ds
N
2 s=0
Z t
1
′
′
≤
ψ̈η (νsN,N )eΓs |Y̺NN (s) − Y̺N ′ (s) | ds
N
2 s=0
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
1
≤
2
Z
t
′
s=0
≤
′
=
A2,N,N
t
≤
≤
≤
′
′
eΓs ψ̈η (νsN,N ){|νsN,N | + |YsN − Y̺NN (s) | + |YsN − Y̺N ′ (s) |} ds
N
t
1 N
1 N′
N
N′
1 + |Ys − Y̺N (s) | + |Ys − Y̺ ′ (s) | ds
e
N
η
η
s=0
2 Z t 1
1 N′
1 N
N
N′
2Γs
e + 1 + |Ys − Y̺N (s) | + |Ys − Y̺ ′ (s) |
ds
N
4 ln η −1 s=0
η
η
Z t 3 N′
3 N
1
2
2
N
N′
2Γs
|Y
−
Y
|Y
|
+
−
Y
e
+
3
+
|
ds,
̺N (s)
̺N ′ (s)
4 ln η −1 s=0
η2 s
η2 s
Z
1 t
′
′
ψ̈η (νsN,N ){((Y̺NN (s) + Zs ) ∨ 0) − ((Y̺NN ′ (s) + Zs ) ∨ 0)}2 ds
2 s=0
Z
′
′
1 t
ψ̈η (νsN,N )|Y̺NN (s) − Y̺N ′ (s) |2 ds
N
2 s=0
Z t
3
′
′
′
′
ψ̈η (νsN,N ){|νsN,N |2 + |YsN − Y̺NN (s) |2 + |YsN − Y̺N ′ (s) |2 } ds
N
2 s=0
Z t 1 N′
1 N
3
2
2
N
N′
1/2
η + |Ys − Y̺N (s) | + |Ys − Y̺ ′ (s) | ds.
N
2 ln η −1 s=0
η
η
1
≤
2 ln η −1
≤
′
33
Z
Γs
′
In the bound on A1,N,N , we have used Young’s inequality, and in the bound
′
on A2,N,N we have used the fact that the support of ψ̈η is contained in
√
[0, η]. Collecting things together, we have that there is a K > 0 such that
K
1
1
1,N,N ′
]≤
+
1+
,
E[At
ln η −1
N η2 N ′ η2
K
1
1
2,N,N ′
1/2
]≤
E[At
+
η +
ln η −1
N η N ′η
for all t ∈ [0, T ]. Thus,
lim
′
N,N →∞
′
E[|νtN,N |] ≤
√
η+
Kσ 2
Kβ 2 η 1/2
+
ln η −1
ln η −1
′
for all t ∈ [0, T ]. Letting η ց 0, we indeed get that limN,N ′ →∞ E[|νtN,N |] = 0.
We thus have that
lim
N,N ′ →∞
′
E[|YtN − YtN |] = 0.
For any p > 1, we also have by interpolation and (10.1) and (10.2) that
q
′
′
N
N′ p
lim
E[|Yt − Yt | ] ≤ lim
E[|YtN − YtN |]E[|YtN − YtN |2p−1 ]
′
′
N,N →∞
N,N →∞
= 0.
34
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
Thus, there is a solution Y of the integral equation
Z t
Z t
p
∗
Γs /2
((Ys + Zs ) ∨ 0) dVs
(Ys + Zs ) ∨ 0 dWs + β
e
Yt = σ
s=0
s=0
def
such that supt∈[0,T ] E[|Yt |p ] < ∞ for all T > 0 and p ≥ 1. Setting Ȳt = Zt + Yt ,
T
we have that Ȳt ∈ p≥1 Lp and that
Z t
Z t
p
Γs /2
∗
e
Ȳt = Zt + σ
(Ȳs ∨ 0) dVs .
Ȳs ∨ 0 dWs + β
s=0
s=0
We claim that Ȳ is nonnegative. For each η > 0 we have that
Z
σ2 t
ψη (Ȳt )χR− (Ȳt ) = ψη (λ◦ )χR− (λ◦ ) +
ψ̈η (Ȳs )χR− (Ȳs )eΓs /2 (Ȳs ∨ 0) ds
2 s=0
Z
β2 t
ψ̈η (Ȳs )χR− (Ȳs )(Ȳs ∨ 0)2 ds + Mt ,
+
2 s=0
where M is a martingale. Taking expectations and then letting η ց 0, we
def
have that E[Ȳt− ] = 0. We finally set λt = e−Γt Ȳt . The claim follows. Proof of Lemma 3.2.
Let λ and λ′ be two solutions of (3.1). Define
def
def
Yt = λt eΓt − Zt and Yt′ = λ′t eΓt − Zt . Since λ and λ′ are assumed to be
nonnegative, Y and Y ′ satisfy
Z t
Z t
p
∗
S
Γs /2
(Ys + Zs ) dVs ,
e
Yt = σ
Ys + Zs dWs + β
Yt′ = σ
def
Set νt =
s=0
t
Z
Yt − Yt′ .
s=0
eΓs /2
p
Ys′ + Zs dWs∗ + β S
s=0
t
Z
s=0
(Ys′ + Zs ) dVs .
For each η > 0,
√
√
|νt | ≤ ψη (νt ) + η = σ 2 A1t + (β S )2 A2t + Mt + η,
where M is a martingale and where
Z t
Z
p
p
1
1 t
2
1
Γs
′
At =
eΓs ds,
ψ̈η (νs )e { Ys + Zs − Ys + Zs } ds ≤
2 s=0
ln η −1 s=0
Z
1 t
η 1/2
2
At =
t.
ψ̈η (νs )νs2 ds ≤
2 s=0
ln η −1
Collecting things together, we have that
Z t
√
1
√
Γs
E[e ] ds .
ηt +
E[|νt |] ≤ η +
ln η −1
s=0
Let η ց 0 to get that Y = Y ′ . The claim follows. DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
35
Let’s next prove the needed macroscopic bound on the λN,n ’s.
For each N ∈ N and n ∈ {1, 2, . . . , N }, define
Z t
N,n def
S
Xs ds,
Γt = αN,n t + βN,n γ
Proof of Lemma 3.4.
s=0
def
ZtN,n =
and let Y
N,n
λN,n,◦ + αN,n λ̄N,n
Z
t
ΓN,n
s
e
s=0
C
ds + βN,n
Z
t
s=0
eΓs dLN
s
satisfy the equation
q
Z t
N,n
ΓN,n
/2
s
e
Yt
= σN,n
YsN,n + ZsN,n dWsn
s=0
S
+ εN βN,n
then
λN,n
t
ΓN,n
t
=e
(YtN,n
+ ZtN,n ).
−2pΓN,n
t
|λN,n
|p ≤ 21 {e
t
N,n
≤ 21 {e−2pΓt
Z
t
s=0
(YsN,n + ZsN,n ) dVs ;
We calculate that
+ |YtN,n + ZtN,n |2p }
+ 22p−1 (|YtN,n |2p + |ZtN,n |2p )}.
From Lemma 10.1, we have that
sup
0≤t≤T
N ∈N
N
N,n
1 X
E[e2pΓt ] and
N
sup
0≤t≤T
N ∈N
n=1
N
1 X
E[|ZtN,n |2p ]
N
n=1
are both finite.
For each N ∈ N and n ∈ {1, 2, . . . , N }, we compute that
Z t
N,n
N,n 2p
2
E[|Yt | ] = p(2p − 1) σN,n
E[|YsN,n |2p−2 eΓs |YsN,n + ZsN,n |] ds
s=0
S
+ ε2N (βN,n
)2
To bound the integrals, we have that
Z
t
s=0
E[|YsN,n |2p−2 |YsN,n
+ ZsN,n |2 ] ds
N,n
|YsN,n |2p−2 eΓs |YsN,n + ZsN,n |
≤
p − 1 N,n 2p
1
1 2pΓN,n
e t +
|Yt | + |YsN,n + ZsN,n |2p
2p
p
2p
≤
p − 1 N,n 2p 22p−1
1 2pΓN,n
e t +
|Yt | +
{|YsN,n |2p + |ZsN,n |2p },
2p
2p
2p
|YsN,n |2p−2 |YsN,n + ZsN,n |2
.
36
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
p − 1 N,n 2p
|Ys | +
p
p − 1 N,n 2p
|Ys | +
≤
p
≤
1 N,n
|Y
+ ZsN,n |2p
p s
22p−1
{|YsN,n |2p + |ZsN,n |2p }.
p
Combining things together, we have that there is a K > 0 such that
2
S
E[|YtN,n |2p ] ≤ K{σN,n
+ ε2N (βN,n
)2 }
Z t
Z
N,n 2p
E[|Ys | ] ds +
×
+
N,n
E[e2pΓs ] ds
s=0
t
s=0
Z
t
s=0
E[|ZsN,n |2p ] ds
for all N ∈ N and n ∈ {1, 2, . . . , N }. Using Assumption 2.3 and averaging
over n, we get the claimed result. 10.3. Proof of Lemma 4.1. Define a homeomorphism Φ of C[0, ∞) as
Z t
Z
def
C p
p
β ḃ (t)λ +
ḃ (t − r){q(r) + αλ̄} dr
Φ(q)(t) =
r=0
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
Z
p
× exp −b (t)λ −
t
b (t − r){q(r) + αλ̄} dr
p
r=0
× π(dp)Λ◦ (dλ)
for all q ∈ C[0, ∞) and t ≥ 0. Note that since b, q and λ are all nonnegative,
Z t
p
p
b (t − r){q(r) + αλ̄} dr ≤ 1.
0 ≤ exp −b (t)λ −
r=0
We can then set up a recursion; we want to solve Q = Φ(Q). Note that there
is a K > 0 such that
Z t
q(r) dr
|Φ(q)(t)| ≤ K
s=0
for all nonnegative q ∈ C[0, ∞).
For any q1 and q2 in C[0, ∞), we have that
Φ(q1 )(t) − Φ(q2 )(t) = Γat (q1 , q2 ) + Γbt (q1 , q2 ),
where
Γat (q1 , q2 )
Z t Z
def
=
s=0
1
θ=0
Z
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
β C ḃp (t − s){q1 (s) − q2 (s)}
DEFAULT CLUSTERING IN LARGE PORTFOLIOS: TYPICAL EVENTS
× exp −bp (t)λ
Z t
p
b (t − r)[{q2 (r) + θ(q1 (r) − q2 (r))} + αλ̄] dr
−
r=0
× π(dp)Λ◦ (dλ) dθ ds,
Γbt (q1 , q2 )
def
=−
37
Z
t
s=0
Z
1
θ=0
Z
β
p̂=(p,λ)∈P̂
p=(α,λ̄,σ,β C ,β S )
C
ḃ (t)λ +
p
Z
t
r=0
ḃp (t − r)
× [{q2 (r) + θ(q1 (r) − q2 (r))} + αλ̄] dr
× {bp (t − s)(q1 (s) − q2 (s))}
× exp −bp (t)λ
−
Z
t
r=0
bp (t − r)[{q2 (r) + θ(q1 (r) − q2 (r))} + αλ̄] dr
× π(dp)Λ◦ (dλ) dθ ds.
Standard techniques from Picard iterations give us the result.
Acknowledgments. R. B. Sowers would like to thank the Departments
of Mathematics and Statistics of Stanford University for their hospitality
in the Spring of 2010 during a sabbatical stay. The authors are grateful to
Thomas Kurtz for some insight into the literature on mean field models, and
to Michael Gordy and participants of the 3rd SIAM Conference on Financial Mathematics and Engineering in San Francisco and the 2010 Annual
INFORMS Meeting in Austin for comments.
REFERENCES
[1] Azizpour, S., Giesecke, K. and Schwenkler, G. (2010). Exploring the sources of
default clustering. Working paper, Stanford Univ.
[2] Bush, N., Hambly, B. M., Haworth, H., Jin, L. and Reisinger, C. (2011).
Stochastic evolution equations in portfolio credit modelling. SIAM J. Financial Math. 2 627–664. MR2836495
[3] Dai Pra, P., Runggaldier, W. J., Sartori, E. and Tolotti, M. (2009). Large
portfolio losses: A dynamic contagion model. Ann. Appl. Probab. 19 347–394.
MR2498681
[4] Dai Pra, P. and Tolotti, M. (2009). Heterogeneous credit portfolios and the
dynamics of the aggregate losses. Stochastic Process. Appl. 119 2913–2944.
MR2554033
38
K. GIESECKE, K. SPILIOPOULOS AND R. B. SOWERS
[5] Das, S., Duffie, D., Kapadia, N. and Saita, L. (2007). Common failings: How
corporate defaults are correlated. J. Finance 62 93–117.
[6] Davis, M. H. A. and Esparragoza-Rodriguez, J. C. (2007). Large portfolio credit
risk modeling. Int. J. Theor. Appl. Finance 10 653–678. MR2341487
[7] Dawson, D. A. and Hochberg, K. J. (1982). Wandering random measures in the
Fleming–Viot model. Ann. Probab. 10 554–580. MR0659528
[8] Deelstra, G. and Delbaen, F. (1994). Existence of solutions of stochastic differential equations related to the Bessel process. Working paper. Dept. Mathematics,
ETH, Zürich, Switzerland.
[9] Dembo, A., Deuschel, J.-D. and Duffie, D. (2004). Large portfolio losses. Finance
Stoch. 8 3–16. MR2022976
[10] Duffie, D., Pan, J. and Singleton, K. (2000). Transform analysis and asset pricing
for affine jump-diffusions. Econometrica 68 1343–1376. MR1793362
[11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and
Convergence. Wiley, New York. MR0838085
[12] Fleming, W. H. and Viot, M. (1979). Some measure-valued Markov processes in
population genetics theory. Indiana Univ. Math. J. 28 817–843. MR0542340
[13] Giesecke, K. and Weber, S. (2006). Credit contagion and aggregate losses.
J. Econom. Dynam. Control 30 741–767. MR2224986
[14] Glasserman, P., Kang, W. and Shahabuddin, P. (2007). Large deviations in
multifactor portfolio credit risk. Math. Finance 17 345–379. MR2332261
[15] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland,
Amsterdam. MR1011252
[16] Jakša Cvitanić, J. M. and Zhang, J. (2012). Law of large numbers for self-exciting
correlated defaults. Stochastic Process. Appl. To appear.
[17] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus,
2nd ed. Graduate Texts in Mathematics 113. Springer, New York. MR1121940
[18] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion.
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences] 293. Springer, Berlin. MR1083357
[19] Royden, H. L. (1988). Real Analysis, 3rd ed. Macmillan Publishing Company, New
York. MR1013117
[20] Sircar, R. and Zariphopoulou, T. (2010). Utility valuation of multi-name credit
derivatives and application to CDOs. Quant. Finance 10 195–208. MR2642963
K. Giesecke
Department of Management Science
and Engineering
Stanford University
Stanford, California 94305-4026
USA
E-mail: [email protected]
K. Spiliopoulos
Division of Applied Mathematics
Brown University
Providence, Rhode Island 029126
USA
E-mail: [email protected]
R. B. Sowers
Department of Mathematics
University of Illinois
at Urbana, Champaign
Urbana, Illinois 61801
USA
E-mail: [email protected]